Suppose C  is a NDCC corresponding to the quadratic polynomial $P(x,y)=Ax^2 + Bxy+Cy^2+Dx+Ey+F$ .

Definition: Let $\Delta = B^2 - 4AC$.
$\Delta$ is called the discriminant of $P$.

The Discriminant Theorem: 

  1. C  is an ellipse if and only if  $\Delta $ < $0$.
  2. C is a parabola if and only if  $\Delta = 0$.
  3. C is an hyperbola if and only if  $\Delta $ > $0$.
Proof: The proof is broken into three steps.
Since 1,2,and 3 are mutually exclusive and exhaustive cases for a NDCC,
the converse statements are also true.